Subgroups — C₄×#88794 #13158987

Order 1728.

1012 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6308	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#88794	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#11722054	C₂	1	1	yes	maximal, characteristic, complemented
6	#11722041	C₂	1	1	yes	maximal, characteristic, complemented
7	#11722032	C₂	1	1	yes	maximal, characteristic
8	C₄×#3754	—	3	1	no	maximal
9	C₄×#3716	—	3	4	no	maximal
10	C₂×#6308	C₂²	1	1	yes	characteristic
11	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄×D₃	—	2	1	no	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
15	#88857	—	2	1	no	—
16	C₄×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
20	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
21	#88794	C₄	1	4	yes	direct product factor
22	C₃²×C₄×D₄	—	3	1	no	—
23	C₂×#3754	—	3	1	no	—
24	C₄×#634	—	3	1	no	—
25	C₄×#634	—	3	4	no	—
26	C₄×#634	—	6	4	no	—
27	C₃×C₄×D₁₂	D₃	1	4	yes	—
28	C₂×#3716	—	3	4	no	—
29	#63659	—	3	1	no	—
30	#63637	—	3	1	no	—
31	#63635	—	3	1	no	—
32	C₄×C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
33	C₃×C₄×C₃⋊C₈	—	3	4	no	—
34	#63573	—	3	4	no	—
35	#63567	—	3	4	no	—
36	#63563	—	3	4	no	—
37	C₂²×C₃²×D₃	—	2	1	no	—
38	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
40	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
42	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
43	#6308	—	2	1	no	—
44	#6308	C₂×C₄	1	2	yes	complemented
45	C₂×C₃²×Dic₃	—	2	1	no	—
46	C₃²×D₁₂	—	2	1	no	—
47	C₃²×D₁₂	C₂×C₄	1	2	yes	—
48	C₃²×C₄×D₃	—	4	2	no	—
49	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
50	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	complemented
51	C₄×#214	—	9	4	no	—
52	C₄×#204	—	9	1	no	—
53	C₂×C₃²×D₄	—	3	1	no	—
54	C₂²×C₃²×C₄	—	6	1	no	—
55	C₂×#634	—	3	1	no	—
56	C₂×#634	—	3	4	no	—
57	C₂×#634	—	6	4	no	—
58	C₂×C₃×D₁₂	C₂×D₃	1	4	yes	—
59	C₂×C₃×C₄×D₃	—	2	4	no	—
60	C₃²×C₄⋊C₄	—	3	1	no	—
61	C₃²×C₂²⋊C₄	—	6	1	no	—
62	C₃²×C₄²	—	2	4	no	—
63	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
64	C₃²×C₄²	C₂×D₃	1	4	yes	—
65	#3754	—	3	4	no	—
66	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
67	C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
68	C₄⋊(C₃⋊Dic₃)	—	6	4	no	—
69	#3750	—	6	8	no	—
70	#3750	—	6	1	no	—
71	#3750	—	6	4	no	—
72	C₄×C₃⋊Dic₃	—	6	8	no	—
73	C₄×C₃⋊Dic₃	—	6	1	no	—
74	C₄×C₃⋊Dic₃	—	6	4	no	—
75	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
76	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
77	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
78	C₃×C₄⋊Dic₃	C₂×D₃	1	4	yes	—
79	C₂×C₃×C₃⋊C₈	—	3	4	no	—
80	C₂×C₃×C₃⋊C₈	—	3	4	no	—
81	#3716	—	3	16	no	—
82	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×D₃	—	2	2	no	—
84	C₂×C₃²×D₃	—	4	1	no	—
85	C₂×C₃²×D₃	—	4	1	no	—
86	C₃³×C₄	—	2	1	no	—
87	C₃³×C₄	SD₁₆	1	2	yes	complemented
88	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₃³×C₄	#41	1	1	yes	characteristic
90	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
91	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
92	C₃²×Dic₃	—	4	1	no	—
93	C₃×C₄×D₄	—	3	4	no	—
94	C₂×#214	—	9	4	no	—
95	C₂×#204	—	9	1	no	—
96	C₄×D₁₂	C₃⋊D₃	1	1	yes	characteristic
97	C₄×Dic₆	—	9	4	no	—
98	C₄×Dic₆	—	9	1	no	—
99	C₄×Dic₆	—	18	4	no	—
100	C₃×C₄×C₈	—	9	1	no	—
101	#832	—	9	4	no	—
102	#821	—	9	1	no	—
103	#817	—	9	1	no	—
104	#816	—	9	1	no	—
105	#810	—	9	4	no	—
106	#808	—	9	4	no	—
107	C₄×C₃⋊C₈	—	3	4	no	—
108	C₂³×C₃²	—	6	1	no	—
109	C₂²×C₃×D₃	—	2	4	no	—
110	C₃²×D₄	—	3	2	no	—
111	C₃²×D₄	—	6	1	no	—
112	C₂×C₃²×C₄	—	2	4	no	—
113	C₂×C₃²×C₄	—	2	4	no	—
114	C₂×C₃²×C₄	—	2	4	no	—
115	C₂×C₃²×C₄	—	6	1	no	—
116	C₂×C₃²×C₄	—	12	2	no	—
117	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
118	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
119	C₂×C₃²×C₄	#67	1	4	yes	—
120	C₂×C₃²×C₄	#67	1	4	yes	—
121	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₃⋊Dic₃	—	6	8	no	—
124	C₂×C₃⋊Dic₃	—	6	1	no	—
125	C₂×C₃⋊Dic₃	—	6	4	no	—
126	C₂×C₃⋊Dic₃	—	6	8	no	—
127	C₂×C₃⋊Dic₃	—	6	1	no	—
128	C₂×C₃⋊Dic₃	—	6	4	no	—
129	#634	—	3	2	no	—
130	#634	—	3	8	no	—
131	#634	—	6	8	no	—
132	#634	—	6	8	no	—
133	#634	—	6	4	no	—
134	#634	—	6	1	no	—
135	C₂×C₃×Dic₃	—	2	4	no	—
136	C₃×D₁₂	—	2	4	no	—
137	C₃×D₁₂	C₄×D₃	1	8	yes	—
138	C₃×C₄×D₃	—	4	8	no	—
139	C₃⋊(C₃⋊C₈)	—	2	1	no	—
140	C₃⋊(C₃⋊C₈)	C₄×D₃	1	2	yes	complemented
141	C₃×C₃⋊C₈	—	3	8	no	—
142	C₃×C₃⋊C₈	—	6	4	no	—
143	C₄×SD₁₆	—	27	1	no	Sylow
144	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
145	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₃³	#135	1	1	yes	characteristic
147	C₃²×D₃	—	4	2	no	—
148	C₂×C₃×D₄	—	3	4	no	—
149	C₂²×C₃×C₄	—	6	4	no	—
150	C₂×D₁₂	C₂×C₃⋊D₃	1	1	yes	characteristic
151	C₂×C₄×D₃	—	2	1	no	—
152	C₂×Dic₆	—	9	4	no	—
153	C₂×Dic₆	—	9	1	no	—
154	C₂×Dic₆	—	18	4	no	—
155	C₂×C₃×C₈	—	9	1	no	—
156	C₂×C₃×C₈	—	9	1	no	—
157	C₃×C₄⋊C₄	—	3	4	no	—
158	C₃×C₂²⋊C₄	—	6	4	no	—
159	C₃×C₄²	—	2	4	no	—
160	C₃×C₄²	D₃²	1	4	yes	—
161	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
162	#214	—	9	16	no	—
163	(C₂×D₃)⋊C₄	—	2	1	no	—
164	C₄⋊Dic₃	—	9	4	no	—
165	C₄⋊Dic₃	—	9	1	no	—
166	C₄⋊Dic₃	—	18	4	no	—
167	C₄⋊Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
168	Dic₃⋊C₄	—	18	8	no	—
169	Dic₃⋊C₄	—	18	4	no	—
170	Dic₃⋊C₄	—	18	1	no	—
171	C₄×Dic₃	—	18	8	no	—
172	C₄×Dic₃	—	18	4	no	—
173	C₄×Dic₃	—	18	1	no	—
174	C₂×C₃⋊C₈	—	3	4	no	—
175	C₂×C₃⋊C₈	—	3	4	no	—
176	#204	—	9	4	no	—
177	C₂²×C₃²	—	2	4	no	—
178	C₂²×C₃²	—	6	2	no	—
179	C₂²×C₃²	—	12	1	no	—
180	C₂²×C₃²	—	12	1	no	—
181	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
182	C₂²×C₃²	C₂×#67	1	4	yes	—
183	C₂×C₃×D₃	—	2	8	no	—
184	C₂×C₃×D₃	—	4	4	no	—
185	C₂×C₃×D₃	—	4	4	no	—
186	C₃²×C₄	—	2	1	no	—
187	C₃²×C₄	—	2	8	no	—
188	C₃²×C₄	—	2	4	no	—
189	C₃²×C₄	—	2	4	no	—
190	C₃²×C₄	—	2	4	no	—
191	C₃²×C₄	—	4	4	no	—
192	C₃²×C₄	—	12	1	no	—
193	C₃²×C₄	#204	1	2	yes	complemented
194	C₃²×C₄	#214	1	8	yes	complemented
195	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
196	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
197	C₃²×C₄	#235	1	4	yes	—
198	C₃²×C₄	#235	1	1	yes	characteristic
199	C₃⋊Dic₃	—	6	16	no	—
200	C₃⋊Dic₃	—	6	2	no	—
201	C₃⋊Dic₃	—	6	8	no	—
202	C₃⋊Dic₃	—	12	8	no	—
203	C₃⋊Dic₃	—	12	1	no	—
204	C₃⋊Dic₃	—	12	4	no	—
205	C₃×Dic₃	—	4	4	no	—
206	C₂×SD₁₆	—	27	1	no	—
207	C₄×Q₈	—	27	1	no	—
208	C₄×D₄	—	3	1	no	—
209	#106	—	27	1	no	—
210	#103	—	27	1	no	—
211	#102	—	27	1	no	—
212	C₄×C₈	—	9	1	no	—
213	C₃³	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₃	—	6	4	no	—
215	C₂²×D₃	—	2	1	no	—
216	C₃×D₄	—	3	8	no	—
217	C₃×D₄	—	6	4	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	6	4	no	—
222	C₂×C₃×C₄	—	12	8	no	—
223	C₂×C₃×C₄	#626	1	4	yes	—
224	C₂×C₃×C₄	#626	1	4	yes	—
225	C₂×C₃×C₄	#638	1	1	yes	characteristic
226	C₂×C₃×C₄	#638	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
228	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
229	C₂×Dic₃	—	2	1	no	—
230	C₂×Dic₃	—	18	8	no	—
231	C₂×Dic₃	—	18	4	no	—
232	C₂×Dic₃	—	18	1	no	—
233	C₂×Dic₃	—	18	8	no	—
234	C₂×Dic₃	—	18	4	no	—
235	C₂×Dic₃	—	18	1	no	—
236	D₁₂	—	2	1	no	—
237	D₁₂	C₄×C₃⋊D₃	1	2	yes	—
238	C₄×D₃	—	4	2	no	—
239	Dic₆	—	9	8	no	—
240	Dic₆	—	9	2	no	—
241	Dic₆	—	18	8	no	—
242	Dic₆	—	18	8	no	—
243	Dic₆	—	18	4	no	—
244	Dic₆	—	18	1	no	—
245	C₃×C₈	—	9	2	no	—
246	C₃×C₈	—	18	1	no	—
247	C₃⋊C₈	—	3	8	no	—
248	C₃⋊C₈	—	6	4	no	—
249	C₂×C₃²	—	2	4	no	—
250	C₂×C₃²	—	2	4	no	—
251	C₂×C₃²	—	2	4	no	—
252	C₂×C₃²	—	12	2	no	—
253	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
254	C₂×C₃²	C₂×#204	1	1	yes	characteristic
255	C₂×C₃²	#904	1	1	yes	characteristic
256	C₂×C₃²	C₄×#67	1	4	yes	—
257	C₂×C₃²	C₂×#214	1	4	yes	—
258	C₂×C₃²	#950	1	4	yes	—
259	C₃×D₃	—	4	8	no	—
260	C₂×Q₈	—	27	1	no	—
261	C₂×D₄	—	3	1	no	—
262	C₂²×C₄	—	6	1	no	—
263	SD₁₆	—	27	4	no	—
264	C₂×C₈	—	9	1	no	—
265	C₂×C₈	—	9	1	no	—
266	C₄⋊C₄	—	3	1	no	—
267	C₄⋊C₄	—	27	1	no	—
268	C₄⋊C₄	—	54	1	no	—
269	C₂²⋊C₄	—	6	1	no	—
270	C₄²	—	54	1	no	—
271	C₄²	D₃×C₃⋊D₃	1	1	yes	characteristic
272	C₂²×C₃	—	2	4	no	—
273	C₂²×C₃	—	6	8	no	—
274	C₂²×C₃	—	12	4	no	—
275	C₂²×C₃	—	12	4	no	—
276	C₂²×C₃	C₂×#626	1	4	yes	—
277	C₂²×C₃	C₂×#638	1	1	yes	characteristic
278	C₂×D₃	—	2	2	no	—
279	C₂×D₃	—	4	1	no	—
280	C₂×D₃	—	4	1	no	—
281	C₃×C₄	—	2	4	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	8	no	—
286	C₃×C₄	—	4	4	no	—
287	C₃×C₄	—	12	4	no	—
288	C₃×C₄	#3716	1	8	yes	complemented
289	C₃×C₄	#3754	1	2	yes	complemented
290	C₃×C₄	#3798	1	4	yes	—
291	C₃×C₄	C₄×D₃²	1	4	yes	—
292	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
293	C₃×C₄	#3828	1	1	yes	characteristic
294	Dic₃	—	4	1	no	—
295	Dic₃	—	18	16	no	—
296	Dic₃	—	18	2	no	—
297	Dic₃	—	18	8	no	—
298	Dic₃	—	36	8	no	—
299	Dic₃	—	36	4	no	—
300	Dic₃	—	36	1	no	—
301	C₃²	—	2	4	no	—
302	C₃²	C₄×#204	1	1	yes	characteristic, complemented
303	C₃²	C₄×#214	1	4	yes	complemented
304	C₂³	—	6	1	no	—
305	Q₈	—	27	2	no	—
306	Q₈	—	54	1	no	—
307	D₄	—	3	2	no	—
308	D₄	—	6	1	no	—
309	C₂×C₄	—	6	1	no	—
310	C₂×C₄	—	12	2	no	—
311	C₂×C₄	—	54	1	no	—
312	C₂×C₄	—	54	1	no	—
313	C₂×C₄	#6291	1	1	yes	characteristic
314	C₂×C₄	#6291	1	1	yes	characteristic
315	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
316	C₈	—	9	2	no	—
317	C₈	—	18	1	no	—
318	C₂×C₃	—	2	4	no	—
319	C₂×C₃	—	2	4	no	—
320	C₂×C₃	—	2	4	no	—
321	C₂×C₃	—	12	8	no	—
322	C₂×C₃	C₂×#3716	1	4	yes	—
323	C₂×C₃	#63829	1	4	yes	—
324	C₂×C₃	C₄×#626	1	4	yes	—
325	C₂×C₃	C₄×#638	1	1	yes	characteristic
326	C₂×C₃	C₂×#3754	1	1	yes	characteristic
327	C₂×C₃	#64159	1	1	yes	characteristic
328	D₃	—	4	2	no	—
329	C₂²	—	6	2	no	—
330	C₂²	—	12	1	no	—
331	C₂²	—	12	1	no	—
332	C₂²	C₂×#6291	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	12	1	no	—
335	C₄	—	54	2	no	—
336	C₄	—	108	1	no	—
337	C₄	#88794	1	2	yes	direct product factor
338	C₄	#89020	1	1	yes	characteristic
339	C₄	C₄×D₃×C₃⋊D₃	1	1	yes	characteristic
340	C₃	—	2	4	no	—
341	C₃	C₄×#3716	1	4	yes	complemented
342	C₃	C₄×#3754	1	1	yes	characteristic, complemented
343	C₂	—	12	2	no	—
344	C₂	C₂×#88794	1	1	yes	characteristic
345	C₂	#11723377	1	1	yes	characteristic
346	C₂	C₄×#6291	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6291
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
Fitting	C₃³×C₄²	C₂²
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₄²	D₃×C₃⋊D₃
hypocenter	C₃³	C₄×SD₁₆
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₄×#6308	C₂	semidirect
4. C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	semidirect
5. #11722054	C₂	semidirect
6. #11722041	C₂	semidirect
21. #88794	C₄	direct (N × H)
32. C₄×C₃⋊(C₃⋊C₈)	D₃	semidirect
44. #6308	C₂×C₄	semidirect
50. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
87. C₃³×C₄	SD₁₆	semidirect
140. C₃⋊(C₃⋊C₈)	C₄×D₃	semidirect
193. C₃²×C₄	#204	semidirect
194. C₃²×C₄	#214	semidirect
213. C₃³	C₄×SD₁₆	semidirect
288. C₃×C₄	#3716	semidirect
289. C₃×C₄	#3754	semidirect
302. C₃²	C₄×#204	semidirect
303. C₃²	C₄×#214	semidirect
337. C₄	#88794	direct (N × H)
341. C₃	C₄×#3716	semidirect
342. C₃	C₄×#3754	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#6308	2
C₃²×C₄×D₁₂	2
C₂×#88794	2
C₃×C₄×C₃⋊(C₃⋊C₈)	2
#11722054	2
#11722041	2
#11722032	2
C₄×#3754	3
C₄×#3716	3

Subgroups — C4×#88794 #13158987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88794 #13158987